This \(1+1\) D (level 2) Carrollian conformal algebra (CCA\(_{1+1}\) ) can be obtained by a Inonu-Wigner contraction of two copies of the relativistic Witt algebra [1]}, thus paving the way to examining the Carrollian conformal field theories systematically in this singular limit. One of the major stumbling blocks in the study of these theories, as opposed to their 2D relativistic parents, is the lack of separation between the left and the right modes. This is because the contraction of the two copies of the Witt algebra to the CCA\(_{1+1}\)  involves a mixing between the chiral and anti-chiral sectors of the parent algebra. So far, this has been the reason why there has been very little use of complex analysis techniques for \(1+1\) D CCFTs. In this paper, by using an intrinsicWe shall not use any `limiting' argument in this work. approach, we propose a solution to this problem by elevating the space coordinate to a complex variable.
